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=
The
variational
formulation
(
12.10
)
then
reads:
For
m
1
,...,M
,
find
(U
m,j
)
r
j
=
0
∈
V
r
m
+
1
L
such that for all
(W
m,i
)
r
m
V
r
m
+
1
L
0
∈
the following holds:
i
=
r
m
r
m
r
m
k
m
2
C
ij
(U
m,j
,W
m,i
)
I
ij
a(U
m,j
,W
m,i
)
+
=
f
m,i
,
i,j
=
0
i,j
=
0
i
=
0
ϕ
i
where
f
m,i
=
(
−
1
)(U
m
−
1
(t
m
−
1
), W
m,i
)
, with
U
0
(t
0
)
=
u
L,
0
∈
V
L
, and for
i, j
=
1
,...,r
m
,
1
−
1
1
−
1
C
ij
=
ϕ
j
ϕ
j
ϕ
i
I
ij
=
t
t
ϕ
j
d
+
(
−
1
)
(
−
1
),
ϕ
j
ϕ
i
d
=
δ
ij
.
The matrices
C
m
and
I
m
,
m
1
,...,M
, are independent of the time step and can be
calculated in a preprocessing step. Their size, however, depends on the correspond-
ing approximation order
r
m
.
Denoting by
M
and
A
the mass and (wavelet compressed) stiffness matrix with
respect to
(
=
·
,
·
)
and
a(
·
,
·
)
,(
12.10
) takes the matrix form
(r
m
+
1
)N
L
Find
u
m
∈ R
=
such that for
m
1
,...,M
C
m
A
u
m
k
2
I
m
(
ϕ
m
M
)
u
m
−
1
,
(12.13)
⊗
M
+
⊗
=
⊗
u
0
=
u
0
,
ϕ
1
where
u
m
and
ϕ
m
denotes the coefficient vector of
U
|
J
m
:=
(
(
−
1
),...,
ϕ
r
m
+
1
(
1
))
∈ R
r
m
+
1
. Furthermore,
u
0
∈ R
N
L
−
is the coefficient vector of
u
L,
0
with respect to the wavelet basis of
V
L
.
For notational simplicity, we consider for the rest of this section a generic time
step, omit the index
m
and write
C
and
I
for the matrices
C
m
and
I
m
, respectively,
u
,
ϕ
for the vectors appearing in (
12.13
), and
r
for the approximation order. Fur-
thermore, we denote the right hand side by
f
=
(
ϕ
m
M
)
u
m
−
1
.
⊗
12.3.2 Solution Algorithm
The system (
12.13
)ofsize
(r
+
1
)N
L
can be reduced to solving
r
+
1 linear systems
QTQ
be the Schur decomposition of the
(r
of size
N
L
. To this end, let
C
=
+
1
)
×
(r
+
1
)
matrix
C
with a unitary matrix
Q
and an upper triangular matrix
T
. Note that
the diagonal of
T
contains the eigenvalues
λ
1
,λ
2
,...,λ
r
+
1
of
C
. Then, multiplying
(
12.13
)by
Q
⊗
I
from the left results in the linear system
T
A
W
k
2
I
(
Q
⊗
(
Q
⊗
⊗
M
+
⊗
=
g
with
w
=
I
)
u
,
g
=
I
)
f
.
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